comments by Diana Fisher about modeling: In a math class we are always trying to associate behavior and structure, except that the structure we traditionally use is a function formula (that is, the formula f(x) = e ^x will produce a behavior that shows exponential growth). Unfortunately, many students do not internalize this formula and its relationship to exponential growth. So I have used generic STELLA diagrams to represent many of the types of behavior we study in Algebra 2, Pre-Calculus, and Calculus, such as, linear behavior, exponential growth and exponential decay, parabolic behavior, sinusoidal behavior (oscillations), logistic behavior (S-shaped), and convergent behavior (goal-seeking). Students seem to be able to remember the diagram better than the equation.
In the modeling course that I taught, I would show some of the generic structures and their behaviors and then, when observing a reference behavior graph for a particular problem, ask what structure might produce such behavior.
What is really interesting is to start putting simple generic structures together, like linear inflow and exponential outflow (or vice versa) and ask students how this will behave. Even the best students have to think hard about this problem because, typically, they have not seen problems of this type before. Actually, I have found some of the students who are NOT typically at the top of the math class more able to think through the possible behavior, because they have a better developed intuitive problem solving sense – which we often do not give students a chance to display in many traditional math classes. By the way, a really useful example of a linear inflow and exponential outflow example is a drug model where the patient is connected to an IV drip and is metabolizing the therapeutic drug at a rate that is proportional to the level of the drug in his system.
In a math class we are always trying to associate behavior and structure, except that the structure we traditionally use is a function formula (that is, the formula f(x) = e ^x will produce a behavior that shows exponential growth). Unfortunately, many students do not internalize this formula and its relationship to exponential growth. So I have used generic STELLA diagrams to represent many of the types of behavior we study in Algebra 2, Pre-Calculus, and Calculus, such as, linear behavior, exponential growth and exponential decay, parabolic behavior, sinusoidal behavior (oscillations), logistic behavior (S-shaped), and convergent behavior (goal-seeking). Students seem to be able to remember the diagram better than the equation.
In the modeling course that I taught, I would show some of the generic structures and their behaviors and then, when observing a reference behavior graph for a particular problem, ask what structure might produce such behavior.
What is really interesting is to start putting simple generic structures together, like linear inflow and exponential outflow (or vice versa) and ask students how this will behave. Even the best students have to think hard about this problem because, typically, they have not seen problems of this type before. Actually, I have found some of the students who are NOT typically at the top of the math class more able to think through the possible behavior, because they have a better developed intuitive problem solving sense – which we often do not give students a chance to display in many traditional math classes. By the way, a really useful example of a linear inflow and exponential outflow example is a drug model where the patient is connected to an IV drip and is metabolizing the therapeutic drug at a rate that is proportional to the level of the drug in his system.